Interpolating asymptotic integration methods for second-order differential equations

Abstract

The problem of asymptotic behaviour at infinity of solutions to second-order differential equation can be reduced via the Liouville transform to that of an equation with almost constant coefficients. In the present paper, we compare various methods of asymptotic integration in application to the reduced equation $u"-(\lambda^2+\varphi(t))u=0$ and interpolate the corresponding results in the case $\operatorname{Re}\lambda>0$, provided that a complex-valued function $\varphi(t)$ is in a certain sense small for large values of the argument.